Bilinear Interpolation | Fast bilinear interpolation
2.11RectangularTensorProductLagrangeSurfacesTensorproductsurfacesaresomeofthesimplestsurfacestoconstruct,buttheyarealsosomeofthemostimportantsurfacesincomputergraphicsandcomputer-aideddesign.Inthetensorproductconstruction,westartwithtworectangulararraysofsize(m+1)×(n+1):oneforthecontrolpoints{Pij}andoneforthenodes{(Sij,tij)},where0≤i≤mand0≤j≤n(seeFigure2.19).Figure2.19.Dataforatensorproductbicubicinterpolant:(a)representsthenodesinthedomainand(b)representsthecontrolpointsintherange.Thenodesl...
2.11 Rectangular Tensor Product Lagrange SurfacesTensor product surfaces are some of the simplest surfaces to construct, but they are also some of the most important surfaces in computer graphics and computer-aided design. In the tensor product construction, we start with two rectangular arrays of size (m + 1) × (n + 1): one for the control points {Pij} and one for the nodes {(Sij,tij)}, where 0 ≤ i ≤ m and 0 ≤ j ≤ n (see Figure 2.19).
Figure 2.19. Data for a tensor product bicubic interpolant: (a) represents the nodes in the domain and (b) represents the control points in the range. The nodes lie on a rectangular grid, but the control points may be in arbitrary positions. The surface P(s,t) must interpolate the control points Pij at the nodes (si, tj)—that is, P(si, tj) = Pij.
The nodes are in special positions because they lie on a rectangular grid in the parameter plane; that is, they lie...